2.3 \(\int -\sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right ) \,dx\)

2.3.1 Mathematica

ClearAll[x] 
integrand = ArcSin[Sqrt[x + 1] - Sqrt[x]]; 
res = Integrate[integrand, x]; 
TeXForm[res]
 

\[ -\frac {(x+1) \left (2 x-2 \sqrt {x+1} \sqrt {x}+1\right )^2 \left (2 \sqrt {\sqrt {x} \sqrt {x+1}-x} \left (-2 x+2 \sqrt {x+1} \sqrt {x}-3\right )+3 \sqrt {-4 x+4 \sqrt {x+1} \sqrt {x}-2} \log \left (2 \sqrt {\sqrt {x} \sqrt {x+1}-x}+\sqrt {-4 x+4 \sqrt {x+1} \sqrt {x}-2}\right )\right )}{8 \sqrt {2} \left (\sqrt {x+1}-\sqrt {x}\right )^3 \left (x-\sqrt {x+1} \sqrt {x}+1\right )^2}-x \sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right ) \]

2.3.2 Rubi

<< Rubi` 
ClearAll[x] 
integrand = ArcSin[Sqrt[x + 1] - Sqrt[x]]; 
res = Int[integrand, x]; 
TeXForm[res]
 

\[ \frac {\text {Subst}\left (\text {Int}\left (\sqrt {-x^2+\sqrt {x^2-1} x+1},x\right ),x,\sqrt {x+1}\right )}{\sqrt {2}}-x \sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right ) \]

2.3.3 Maple

restart; 
integrand:=arcsin(sqrt(x+1)-sqrt(x)); 
res:=int(integrand,x); 
latex(res)
 

\[ -{\frac {1}{16} \left ( \arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \left ( \tan \left ( {\frac {1}{2}\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) } \right ) \right ) ^{8}+2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+ \sqrt {x} \right ) \right ) \right ) ^{6}-2\, \left ( \tan \left ( 1/2\, \arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{7}+ 18\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{4}-6\, \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{5}+2\,\arcsin \left ( -\sqrt {x+1}+\sqrt { x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{2}+6\, \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{3}+\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) +2\,\tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) \left ( 1+ \left ( \tan \left ( {\frac {1}{2}\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) } \right ) \right ) ^{2} \right ) ^{-2} \left ( \tan \left ( { \frac {1}{2}\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) } \right ) \right ) ^{-2}} \]

2.3.4 Fricas

set output tex off 
setSimplifyDenomsFlag(true) 
ii:=integrate(asin(sqrt(x+1)-sqrt(x)),x) 
latex(ii)
 

\[ {{{\left ( {3 \ {\sqrt {{x+1}}}}+{\sqrt {x}} \right )} \ {\sqrt {{{2 \ {\sqrt {x}} \ {\sqrt {{x+1}}}} -{2 \ x}}}}}+{{\left ( {8 \ x}+3 \right )} \ {\arcsin \left ( {{{\sqrt {{x+1}}} -{\sqrt {x}}}} \right )}}} \over 8 \]

2.3.5 Maxima

ii : integrate(asin(sqrt(x+1)-sqrt(x)),x); 
tex(ii);
 

This reslult is wrong. bug.

\[ {{\pi \,x}\over {2}} \]

2.3.6 XCAS

ii :=  integrate(asin(sqrt(x+1)-sqrt(x)),x); 
 
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [92.1017843988] 
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [53.1277311612] 
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [5.38357630698] 
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [6.79369851155] 
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [84.3561567818] 
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [77.6493344628] 
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [72.519035968] 
Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong. 
The choice was done assuming [t_nostep]=[0] 
Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong. 
The choice was done assuming [t_nostep]=[0] 
Algebraic extensions not allowed in a rootof 
Algebraic extensions not allowed in a rootof 
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [69.9232513234] 
 
latex(ii)
 

\[ \text {did not solve} \]

2.3.7 Sympy

>python 
Python 3.7.3 (default, Mar 27 2019, 22:11:17) 
[GCC 7.3.0] :: Anaconda, Inc. on linux 
 
from sympy import * 
x = symbols('x') 
ii=integrate(asin(sqrt(x+1)-sqrt(x)),x); 
latex(ii)
 

\[ \text {did not solve} \]

2.3.8 MuPad

evalin(symengine,'int(asin(sqrt(x+1)-sqrt(x)),x)')
 

\[ \text {did not solve} \]